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Homogenisation for Maxwell and Friends

Andreas Buchinger, Sebastian Franz, Nathanael Skrepek, Marcus Waurick

TL;DR

This work establishes a rigorous operator-theoretic framework for homogenisation of evolutionary equations under nonlocal $H$-convergence (Schur topology) in the sense of Picard. By exploiting a canonical decomposition $\mathcal{H} = \ker(A) \oplus \operatorname{ran}(A)$ and a compactness condition, the authors prove that the $\operatorname{ran}(A)$-component of the solution operator converges strongly while the $\ker(A)$-component converges only weakly, for coefficient sequences $M_n$ converging in the nonlocal topology to $M$. They derive both local and nonlocal homogenisation results, present a suite of analytic and numerical examples (including Maxwell-type systems with memory effects), and demonstrate how memory terms arise in the homogenised limits. The numerical simulations, implemented in the SOFE framework with DG in time and CG in space, corroborate the predicted convergence behavior and illustrate strong-weak dichotomies across multiple problem classes. This work advances the practical computability and understanding of homogenisation for mixed-type and nonlocal coefficient problems in mathematical physics.

Abstract

We refine the understanding of continuous dependence on coefficients of solution operators under the nonlocal $H$-topology viz Schur topology in the setting of evolutionary equations in the sense of Picard. We show that certain components of the solution operators converge strongly. The weak convergence behaviour known from homogenisation problems for ordinary differential equations is recovered on the other solution operator components. The results are underpinned by a rich class of examples that, in turn, are also treated numerically, suggesting a certain sharpness of the theoretical findings. Analytic treatment of an example that proves this sharpness is provided too. Even though all the considered examples contain local coefficients, the main theorems and structural insights are of operator-theoretic nature and, thus, also applicable to nonlocal coefficients. The main advantage of the problem class considered is that they contain mixtures of type, potentially highly oscillating between different types of PDEs; a prototype can be found in Maxwell's equations highly oscillating between the classical equations and corresponding eddy current approximations.

Homogenisation for Maxwell and Friends

TL;DR

This work establishes a rigorous operator-theoretic framework for homogenisation of evolutionary equations under nonlocal -convergence (Schur topology) in the sense of Picard. By exploiting a canonical decomposition and a compactness condition, the authors prove that the -component of the solution operator converges strongly while the -component converges only weakly, for coefficient sequences converging in the nonlocal topology to . They derive both local and nonlocal homogenisation results, present a suite of analytic and numerical examples (including Maxwell-type systems with memory effects), and demonstrate how memory terms arise in the homogenised limits. The numerical simulations, implemented in the SOFE framework with DG in time and CG in space, corroborate the predicted convergence behavior and illustrate strong-weak dichotomies across multiple problem classes. This work advances the practical computability and understanding of homogenisation for mixed-type and nonlocal coefficient problems in mathematical physics.

Abstract

We refine the understanding of continuous dependence on coefficients of solution operators under the nonlocal -topology viz Schur topology in the setting of evolutionary equations in the sense of Picard. We show that certain components of the solution operators converge strongly. The weak convergence behaviour known from homogenisation problems for ordinary differential equations is recovered on the other solution operator components. The results are underpinned by a rich class of examples that, in turn, are also treated numerically, suggesting a certain sharpness of the theoretical findings. Analytic treatment of an example that proves this sharpness is provided too. Even though all the considered examples contain local coefficients, the main theorems and structural insights are of operator-theoretic nature and, thus, also applicable to nonlocal coefficients. The main advantage of the problem class considered is that they contain mixtures of type, potentially highly oscillating between different types of PDEs; a prototype can be found in Maxwell's equations highly oscillating between the classical equations and corresponding eddy current approximations.
Paper Structure (21 sections, 15 theorems, 69 equations, 11 figures)

This paper contains 21 sections, 15 theorems, 69 equations, 11 figures.

Table of Contents

  1. Introduction
  2. The Setting: Evolutionary Equations
  3. Homogenisation Theory and Theorems
  4. Local $\mathrm{H}$-Convergence
  5. Nonlocal $\mathrm{H}$-Convergence
  6. Homogenisation Theory for Evolutionary Equations
  7. Additional Properties of (Non-)Local $\mathrm{H}$-Convergence
  8. Nonlocal $\mathrm{H}$-Convergence of the Inverse Sequence
  9. A Pasting Theorem
  10. Independence of Boundary Conditions
  11. $\mathrm{H}$-convergence in 2D
  12. Examples
  13. The Examples
  14. Homogenisation limits of the examples
  15. Numerical Simulations
  16. ...and 6 more sections

Key Result

Theorem 2.2

Let $A\colon \mathop{\mathrm{dom}}\nolimits(A)\subseteq \mathcal{H}\to \mathcal{H}$ be skew-selfadjoint, $\nu\in \mathbb{R}$, $M\in \mathcal{M}(\mathcal{H},\nu)$. Assume there exists $c>0$ such that Then, the operator is closable. The closure is continuously invertible, $\mathcal{S}_\nu\coloneqq \overline{\mathcal{B}_\nu}^{-1}$. Moreover, $\lVert\mathcal{S}_{\nu}\rVert \leq 1/c$.

Figures (11)

  • Figure 1: Graph of $\mathds{1}_{O_{n}}$ for $n=1$
  • Figure 2: Solutions of \ref{['ex:ode']} for $n\in\{2,4,8,1024\}$ and for the homogenised problem
  • Figure 3: Results investigating weak convergence for \ref{['ex:ode']}
  • Figure 4: Solutions of \ref{['ex:pde-without-compactness']}\ref{['item:1D-glueing-example']} for $n\in\{2,4,8,1024\}$ from top left to bottom right.
  • Figure 5: Results investigating weak convergence for \ref{['ex:pde-without-compactness']}\ref{['item:1D-glueing-example']}
  • ...and 6 more figures

Theorems & Definitions (39)

  • definition 2.1
  • Theorem 2.2: Picard's Theorem, SeTrWa22
  • remark 2.3
  • example 2.4: Maxwell's Equations
  • definition 3.1: Ta09
  • Theorem 3.2: Ta09
  • Theorem 3.3: CiDo99
  • remark 3.4
  • definition 3.5: cf. Wa18
  • Theorem 3.6: cf. Wa18
  • ...and 29 more